Download Complex 2.3

Mathematical Investigations IV
Name:
Mathematical Investigations IV
Complex Concepts-Making “i” contact
Products, Quotients, DeMoivre’s Theorem
What happens when we multiply two complex numbers? We have observed a relationship
between the radii and angles of the two complex numbers and the resulting product. Let’s prove
this relationship in the general case:
(r1 cis  ) (r2 cis )  r1 r2 cis(   )
(r1 cos  + i r1 sin ) (r2 cos  + i  r2 sin  )
Write (r1 cis )·(r2 cis ) in polar form.
Multiply out the terms in the parentheses.
_____________________________________
Group the real terms and
the imaginary terms together.
_____________________________________
Simplify, using trig identities.
_____________________________________
Rewrite in cis form
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Next, let’s divide one complex number by another and prove the analogous relationship between
the radii and angles of the complex numbers and their quotient:
r1 cis  r1
 cis(   )
r2 cis  r2
Write
r1 cis 
in polar form
r2 cis 
r1 cos  + i r1 sin 
r2 cos  + i r2 sin 
Rationalize the denominator
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Multiply everything out
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Group the real and imaginary terms
________________________________________
Use trig identities to simplify
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Rewrite in cis form
________________________________________
Complex 2.1
Rev F06
Mathematical Investigations IV
Name:
1.
Fill in the blank.


6 cis =
4
5
cis 40 cis 30 = cis _____
3cis
cis
2 
cis  
 3 
= cis_____
 
cis  
12
cis(70)
cis235 = cis ______
cis (120)
(5 cis 20)( ___ cis ____) = 10 cis 70
(4 cis 50)(___ cis ____) = -4 cis 50
2.
For z = r cis , find the following in terms of r and 
z2 = _______________
z3 = _______________
z15 = _______________
zn = _______________
Generalizing from our results from zn, we state DeMoivre’s Theorem:
If z = r cis , then zn = rn cis (n)
3.
Use DeMoivre’s Theorem to simplify the following:
(2 cis 12°)6 =________________
(4 cis 23°)3 = _______________
(cis 35 )6 (cis 20 )4
cis 25
= __________
(4 cis 16°)3 (2 cis (–10°))3 = ____________
Complex 2.2
Rev F06
Mathematical Investigations IV
Name:
(((cis 5°)2 cis 10°)2 cis 20°)2 = ___________
4.
Let z = 2 cis 45°. Find z2, z3, z4, and z5 in polar form. Plot z and each of these powers on
an Argand diagram.
5.
Let z = cis15°. Plot z, z2, z3, ... , z7 on an Argand diagram. Also locate z10, z30, and
z100. z and all of the powers listed above are all solutions to the equation zn = 1 for some
values of n. Find this value of n.
Complex 2.3
Rev F06